Finite Mathematik Beispiele

$w - 2 x - y - 3 z = 0$ , $w + x - y = 3$ , $5 w + 5 x + z = 5$ , $5 x - 2 y + z = 9$

Schritt 1

Write the system as a matrix.

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ 1 & 1 & - 1 & 0 & 3 \\ 5 & 5 & 0 & 1 & 5 \\ 0 & 5 & - 2 & 1 & 9 \end{array}]$

Schritt 2

Ermittele die normierte Zeilenstufenform.

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Schritt 2.1

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

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Schritt 2.1.1

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ 1 - 1 & 1 + 2 & - 1 + 1 & 0 + 3 & 3 - 0 \\ 5 & 5 & 0 & 1 & 5 \\ 0 & 5 & - 2 & 1 & 9 \end{array}]$

Schritt 2.1.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ 0 & 3 & 0 & 3 & 3 \\ 5 & 5 & 0 & 1 & 5 \\ 0 & 5 & - 2 & 1 & 9 \end{array}]$

Schritt 2.2

Perform the row operation $R_{3} = R_{3} - 5 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Schritt 2.2.1

Perform the row operation $R_{3} = R_{3} - 5 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ 0 & 3 & 0 & 3 & 3 \\ 5 - 5 \cdot 1 & 5 - 5 \cdot - 2 & 0 - 5 \cdot - 1 & 1 - 5 \cdot - 3 & 5 - 5 \cdot 0 \\ 0 & 5 & - 2 & 1 & 9 \end{array}]$

Schritt 2.2.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ 0 & 3 & 0 & 3 & 3 \\ 0 & 15 & 5 & 16 & 5 \\ 0 & 5 & - 2 & 1 & 9 \end{array}]$

Schritt 2.3

Multiply each element of $R_{2}$ by $\frac{1}{3}$ to make the entry at $2, 2$ a $1$ .

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Schritt 2.3.1

Multiply each element of $R_{2}$ by $\frac{1}{3}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ \frac{0}{3} & \frac{3}{3} & \frac{0}{3} & \frac{3}{3} & \frac{3}{3} \\ 0 & 15 & 5 & 16 & 5 \\ 0 & 5 & - 2 & 1 & 9 \end{array}]$

Schritt 2.3.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 15 & 5 & 16 & 5 \\ 0 & 5 & - 2 & 1 & 9 \end{array}]$

Schritt 2.4

Perform the row operation $R_{3} = R_{3} - 15 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Schritt 2.4.1

Perform the row operation $R_{3} = R_{3} - 15 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 - 15 \cdot 0 & 15 - 15 \cdot 1 & 5 - 15 \cdot 0 & 16 - 15 \cdot 1 & 5 - 15 \cdot 1 \\ 0 & 5 & - 2 & 1 & 9 \end{array}]$

Schritt 2.4.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 5 & 1 & - 10 \\ 0 & 5 & - 2 & 1 & 9 \end{array}]$

Schritt 2.5

Perform the row operation $R_{4} = R_{4} - 5 R_{2}$ to make the entry at $4, 2$ a $0$ .

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Schritt 2.5.1

Perform the row operation $R_{4} = R_{4} - 5 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 5 & 1 & - 10 \\ 0 - 5 \cdot 0 & 5 - 5 \cdot 1 & - 2 - 5 \cdot 0 & 1 - 5 \cdot 1 & 9 - 5 \cdot 1 \end{array}]$

Schritt 2.5.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 5 & 1 & - 10 \\ 0 & 0 & - 2 & - 4 & 4 \end{array}]$

Schritt 2.6

Multiply each element of $R_{3}$ by $\frac{1}{5}$ to make the entry at $3, 3$ a $1$ .

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Schritt 2.6.1

Multiply each element of $R_{3}$ by $\frac{1}{5}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ \frac{0}{5} & \frac{0}{5} & \frac{5}{5} & \frac{1}{5} & \frac{- 10}{5} \\ 0 & 0 & - 2 & - 4 & 4 \end{array}]$

Schritt 2.6.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & \frac{1}{5} & - 2 \\ 0 & 0 & - 2 & - 4 & 4 \end{array}]$

Schritt 2.7

Perform the row operation $R_{4} = R_{4} + 2 R_{3}$ to make the entry at $4, 3$ a $0$ .

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Schritt 2.7.1

Perform the row operation $R_{4} = R_{4} + 2 R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & \frac{1}{5} & - 2 \\ 0 + 2 \cdot 0 & 0 + 2 \cdot 0 & - 2 + 2 \cdot 1 & - 4 + 2 (\frac{1}{5}) & 4 + 2 \cdot - 2 \end{array}]$

Schritt 2.7.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & \frac{1}{5} & - 2 \\ 0 & 0 & 0 & - \frac{18}{5} & 0 \end{array}]$

Schritt 2.8

Multiply each element of $R_{4}$ by $- \frac{5}{18}$ to make the entry at $4, 4$ a $1$ .

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Schritt 2.8.1

Multiply each element of $R_{4}$ by $- \frac{5}{18}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & \frac{1}{5} & - 2 \\ - \frac{5}{18} \cdot 0 & - \frac{5}{18} \cdot 0 & - \frac{5}{18} \cdot 0 & - \frac{5}{18} (- \frac{18}{5}) & - \frac{5}{18} \cdot 0 \end{array}]$

Schritt 2.8.2

Vereinfache $R_{4}$ .

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & \frac{1}{5} & - 2 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 2.9

Perform the row operation $R_{3} = R_{3} - \frac{1}{5} R_{4}$ to make the entry at $3, 4$ a $0$ .

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Schritt 2.9.1

Perform the row operation $R_{3} = R_{3} - \frac{1}{5} R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 - \frac{1}{5} \cdot 0 & 0 - \frac{1}{5} \cdot 0 & 1 - \frac{1}{5} \cdot 0 & \frac{1}{5} - \frac{1}{5} \cdot 1 & - 2 - \frac{1}{5} \cdot 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 2.9.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 & - 2 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 2.10

Perform the row operation $R_{2} = R_{2} - R_{4}$ to make the entry at $2, 4$ a $0$ .

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Schritt 2.10.1

Perform the row operation $R_{2} = R_{2} - R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ 0 - 0 & 1 - 0 & 0 - 0 & 1 - 1 & 1 - 0 \\ 0 & 0 & 1 & 0 & - 2 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 2.10.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & - 1 & - 3 & 0 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & - 2 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 2.11

Perform the row operation $R_{1} = R_{1} + 3 R_{4}$ to make the entry at $1, 4$ a $0$ .

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Schritt 2.11.1

Perform the row operation $R_{1} = R_{1} + 3 R_{4}$ to make the entry at $1, 4$ a $0$ .

$[\begin{array}{c} 1 + 3 \cdot 0 & - 2 + 3 \cdot 0 & - 1 + 3 \cdot 0 & - 3 + 3 \cdot 1 & 0 + 3 \cdot 0 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & - 2 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 2.11.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & - 2 & - 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & - 2 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 2.12

Perform the row operation $R_{1} = R_{1} + R_{3}$ to make the entry at $1, 3$ a $0$ .

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Schritt 2.12.1

Perform the row operation $R_{1} = R_{1} + R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 + 0 & - 2 + 0 & - 1 + 1 \cdot 1 & 0 + 0 & 0 - 2 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & - 2 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 2.12.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & - 2 & 0 & 0 & - 2 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & - 2 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 2.13

Perform the row operation $R_{1} = R_{1} + 2 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Schritt 2.13.1

Perform the row operation $R_{1} = R_{1} + 2 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + 2 \cdot 0 & - 2 + 2 \cdot 1 & 0 + 2 \cdot 0 & 0 + 2 \cdot 0 & - 2 + 2 \cdot 1 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & - 2 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 2.13.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & - 2 \\ 0 & 0 & 0 & 1 & 0 \end{array}]$

Schritt 3

Use the result matrix to declare the final solution to the system of equations.

$w = 0$

$x = 1$

$y = - 2$

$z = 0$

Schritt 4

The solution is the set of ordered pairs that make the system true.

$(0, 1, - 2, 0)$